[Math] How to solve this exponential decay problem

exponential functionordinary differential equations

This is a problem from page 44 of Edwards & Penneys' Elementary DE Problems, Question #41:

Suppose that a mineral body formed in the ancient cataclysm originally
containing the uranium isotope $^{238}U$ (which has a half-life of
$4.51 * 10^9$ years) but no lead, the end product of the radioactive
decay of $^{238}U$. If today the ratio of $^{238}U$ to lead atoms in
the mineral body is $0.9$, when did the cataclysm occur?

So far I've tried modelling the amount of uranium, but it depends on how much lead there is now. Like this:

$U(t)=U_0e^{\frac{-ln2}{\lambda}t}$

$0.9L=U_0e^{\frac{-ln2}{\lambda}t}$

$t=\frac{-\lambda(\frac{0.9L}{U_0})}{ln2}$

What have I missed?

Best Answer

Here is one approach.

  • Half-life $T = 4.51 \times 10^9$ years
  • Ratio $\dfrac{\mbox{U-238}}{\mbox{Lead}} = 0.9 = \dfrac{9}{10}$

  • This means $\dfrac{\mbox{U-238 atoms}}{\mbox{Original U-238 atoms}}= \dfrac{\dfrac{9}{10}}{0.9 + 1} = \dfrac{9}{19} = \dfrac{N}{N_0}$

  • $\dfrac{N}{N_0} = \left(\dfrac{1}{2}\right)^{\dfrac{t}{T}}$

  • $t = \dfrac{T \ln \left(\dfrac{N}{N_0}\right)}{\ln\left(\dfrac{1}{2}\right)}$
  • $t = \dfrac{4.51 \times 10^9 ~\ln\left(\dfrac{9}{19}\right)}{\ln\left(\dfrac{1}{2}\right)} $
  • $t = 4.86179 \times 10^9$ years
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